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Linear Regression Formula

Linear regression is the most basic and commonly used predictive analysis. One variable is considered to be an explanatory variable, and the other is considered to be a dependent variable. For example, a modeler might want to relate the weights of individuals to their heights using a linear regression model.

There are several linear regression analyses available to the researcher.

Simple linear regression

  • One dependent variable (interval or ratio)
  • One independent variable (interval or ratio or dichotomous)

Multiple linear regression

  • One dependent variable (interval or ratio)
  • Two or more independent variables (interval or ratio or dichotomous)

Logistic regression

  • One dependent variable (binary)
  • Two or more independent variable(s) (interval or ratio or dichotomous)

Ordinal regression

  • One dependent variable (ordinal)
  • One or more independent variable(s) (nominal or dichotomous)

Multinomial regression

  • One dependent variable (nominal)
  • One or more independent variable(s) (interval or ratio or dichotomous)

Discriminant analysis

  • One dependent variable (nominal)
  • One or more independent variable(s) (interval or ratio)

Formula for linear regression equation is given by:

\[\large y=a+bx\]

a and b are given by the following formulas:

\[\large b\left(slope\right)=\frac{n\sum xy-\left(\sum x\right)\left(\sum y\right)}{n\sum x^{2}-\left(\sum x\right)^{2}}\]

\[\large a\left(intercept\right)=\frac{n\sum y-b\left(\sum x\right)}{n}\]

Where,
x and y are two variables on regression line.
b = Slope of the line.
a = y-intercept of the line.
x = Values of first data set.
y = Values of second data set.

Solved Examples

Question: Find linear regression equation for the following two sets of data:

  x  2  4  6 8
y 3 7 5  10

Solution: 

Construct the following table:

x y x2 xy
2 3 4 6
4 7 16 28
6 5 36 30
8 10 64 80
 $\sum x$ = 20  $\sum y$ = 25  $\sum x^{2}$ = 120  $\sum xy$ = 144

$b$ = $\frac{n\sum xy-(\sum x)(\sum y)}{n\sum x^{2}-(\sum x)^{2}}$

$b$ = $\frac{4 \times 144 – 20 \times 25}{4 \times 120 – 400}$

b = 0.95

$a$ = $\frac{\sum y-b \left (\sum x \right )}{n}$

$a$ = $\frac{26 – 0.95 \times 20}{4}$

a = 1.5

Linear regression is given by:

y = a + bx

y = 1.5 + 0.95 x